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19.86 (2018)

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Suppose that a finite group $G$ has a $\sigma_i$-Hall subgroup for every $i \in I$. Suppose that a subgroup $A \leqslant G$ is such that $A \cap H$ is a $\sigma_i$-Hall subgroup of $A$ for every $i \in I$ and every $\sigma_i$-Hall subgroup $H$ of $G$ (see 19.84). Is it true that then $A$ is $\sigma$-subnormal in $G$?

Contributor: A. N. Skiba

Progress

An affirmative answer is known if $\sigma = \{\{2\}, \{3\}, \dots\}$ and if $G$ is $\sigma$-soluble.

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